Cantor's 1891 Diagonal Argument: Three Refutations Revisited
2025
Abstract
The first theorem of Cantor's 1891 paper introduced the diagonal method by a specific example; namely, as an argument that any list of the set of real numbers is necessarily incomplete and concluding the set of real numbers is not denumerable. The present paper argues that the diagonal method is not a valid form of argument (i.e., not a valid proof schema). Any of the three independent refutations presented here suffice to prove that Cantor's Diagonal Argument (CDA) cannot logically lead to the conclusion(s) alleged for it. The first of the three refutations is applicable to any alleged proof by diagonal method, including its application to non-numeric sets. "And as of 'no-go theorems'[:] … One always has to take the assumptions into consideration, just as the small print in a contract."-Gerard t'Hooft (1999 Nobelist in Physics) [1] I.
References (39)
- Post Office Box 2097, Boulder Creek, CA 95006 Email: dmcgoveran@gmail.com .0000000000000000... .1111111111111111... .1000000000000000... .0111111111111111... .0100000000000000... .1011111111111111... .1100000000000000... .0011111111111111... .0010000000000000... .1101111111111111... .0110000000000000... .1001111111111111... .1010000000000000... .0101111111111111... .1110000000000000... .0001111111111111... .0001000000000000... .1110111111111111... .0011000000000000... .1100111111111111... .0101000000000000... .1010111111111111... .0111000000000000... .1000111111111111... .1001000000000000... .0110111111111111... .1011000000000000... .0100111111111111... .1101000000000000... .0010111111111111... .1111000000000000... .0000111111111111...
- ⋮ ⋮ Preprint -September 16, 2025
- Post Office Box 2097, Boulder Creek, CA 95006 Email: dmcgoveran@gmail.com REFERENCES
- t'Hooft, G.(2016) The Cellular Automaton Interpretation of Quantum Mechanics. Springer Open, Cham Heidelberg New York Dordrecht London. Quote from page 8, section 1.1.
- Cantor, G.(1891) "Ueber eine elementare Frage der Mannigfaltigkeitslehre." ("On an Elementary Question of the Theory of Manifoldness") Jahresbericht der Deutsche Mathematiker-Vereinigung 1890-1891. Volume 1: pp. 75-78.
- Cantor, G.(1874) "On a Property of the Collection of All Real Algebraic Numbers." in Ewald, William B., (ed.), From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford University Press. © 1996. Vol.2, pp. 840-843.
- Du Bois-Reymond, Paul.(1875) Über asymptotische Werte, infinitäre Approximationen und infinitäre Auflösungen von Gleichungen. [On asymptotic values, infinitary approximations and infinitary solutions of equations.] Mathematishce Annalen Volume 8: pp. 363-414.
- McCarty, David Charles.(2004) "David Hilbert and Paul du Bois-Reymond: Limits and Ideals." in One Hundred Years of Russell ś Paradox: Mathematics, Logic, Philosophy. Godehard Link, editor. De Gruyter. Pp. 517-532f.
- Cantor, G.(1883) "Foundations of a General Theory of Manifolds: A Mathematico- Philosophical Investigation into the Theory of the Infinite." (Grundlagen einer allgemeinen Mannigfaltigkeitslehre) in Ewald, William B., (ed.), From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford University Press. © 1996. Volume 2, pp. 879-920.
- Cantor, G.(1895) "Beiträge zur Begründung der transfiniten Mengenlehre (1)." ("Contributions to the Founding of the Theory of Transfinite Numbers"). Mathematische Annalen. Volume 46(4): pp. 481-512. Also Dover 1955. p.107.
- Robinson, A.(1964) Formalism 64, Proceedings of the International Congress for Logic, Methodology and Philosophy of Science, Jerusalem (1964), p.230.
- Brouwer, L.E.J.(1975) Collected Works, Vol 1: Philosophy and Foundations of Mathematics. North-Holland, 1975.
- Chaitin, G.(2005) "Meta-Math! : The Quest for Omega." Knopf Doubleday Publishing Group. Digital edition: Nov. 26, 2008.
- Carey, Patrick H.(2005) Beyond Infinity: Georg Cantor and Leopold Kronecker's Dispute over Transfinite Numbers. Boston College Thesis, May 2005. http://hdl.handle.ne/2345/481
- "Controversy over Cantor". https://en.wikipedia.org/wiki/Controversy_over_Cantor%27s_theory
- McGoveran, D.(Dec. 23, 2022) " Interval Arguments: Two Refutations of Cantor's 1874 and 1878." https://www.academia.edu/93528167/Interval_Arguments_Two_Refutations_of_Cantors_18 74_and_1878_Arguments
- Wittgenstein, L.(1937-1944) Remarks on the Foundations of Mathematics. (G. E. M. Anscombe, Trans.). Blackwell, (c) 1964.
- "Nicolas Bourbaki", https://en.wikipedia.org/wiki/Nicolas_Bourbaki, Feb. 2022.
- Bourbaki, N. (1939-1957) Theory of Sets. Éditions Hermann. Paris, France. January 1, 1968. ASIN: B0006C5TOY
- Simmons, Keith.(1993) Universality and the Liar: An essay on truth and the diagonal argument. Cambridge University Press, (digital) 2004.
- "Diagonal Argument". https://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument
- Hilbert, D.(1925) "On the Infinite." Originally "Uber das Unendlich." Mathematische Annalen 95 (1926) English translation by Putnam, E. and Massey, G. in Putnam, H. and Benaceraf, P. (eds.) Philosophy of Mathematics: Selected Readings, 2nd Edition. Cambridge University.
- Date, C. J.(2003) On Cantor and the Transfinite. ©2003 C. J. Date. Technics Publications. ISBN-13: 978-1634623278
- Sayan, Erdinç.(Aug. 15, 2019) "Contra Cantor: How to Count the Uncountably Infinite." https://www.academia.edu/37229455/CONTRA_CANTOR_HOW_TO_COUNT_THE_UN COUNTABLY_INFINITE_ (Erdinç Sayan argues that CDA is invalid, claiming that paradoxical value of the intersection term of the anti-diagonal with itself implied a neither 0 nor 1 (m or w). Although interesting, we find the argument specious since the logical flaw cited here resolves the paradox and precludes the anti-diagonal from spanning its appearance in L.)
- McGoveran, D.(2025) "The Diagonal Method: Refutations of Generalizations and Variations". (formerly "Variations on Diagonal Arguments", version 1:2021-10-02 and version 2:2021-12-10). © 2022-2025 (in progress). Preprint -September 16, 2025
- Dedekind, R.(1872) "Continuity and irrational numbers." in Ewald, William B., (ed.), From Kant to Hilbert: A Source Book in the Foundations of Mathematics, 2 vols. Oxford University Press. © 1996. Vol. 1, pp. 765-78.
- McGoveran, D.(2022) "Cantor's 1891 Diagonal Argument: Three Refutations." © 2016- 2022: https://www.academia.edu/70865226/Cantors_1891_Diagonal_Argument_Three_Refutations (The present paper, based on material already in earliest versions (v1-v7) under various titles, including "Diagonalization" as distributed in confidence to select reviewers, from January 2016 through January 2022. © February, 2022. Note: Some early versions of paper mistakenly cited "1874" instead of "1891" when referring to the relevant paper and theorem by Cantor. Mea culpa!)
- McGoveran, D.(2024) "Cantor's 1891 Diagonal Argument: Comments on Cantor's Assumptions." © 2024. https://www.academia.edu/124456210/Cantors_1891_Diagonal_Argument_Comments_on_ Cantors_Assumptions
- McGoveran, D.(2025) "On An Uncountably Infinite Proper Subset of the Rationals." © 2024-2025. https://www.academia.edu/127636319/On_An_Uncountably_Infinite_Proper_Subset_of_the _Rationals
- Cantor, G.(1875). Letter to Richard Dedekind, dated Nov.1875.
- McGoveran, D.(2025) "The Pervasive Impact of the Diagonal Method". (Portions of this paper appeared in even the earliest versions of "Cantor's 1891 Diagonal Argument".) © 2025 (in progress).
- "Liar paradox." Wikipedia. https://en.wikipedia.org/wiki/Liar_paradox
- "Russell's paradox." Wikipedia. https://en.wikipedia.org/wiki/Russell%27s_paradox
- Mückenheim, Wolfgang.(2022) "Transfinity." https://www.hs-augsburg.de/~mueckenh/ Transfinity/Transfinity/pdf
- Kuan Peng.(2016) "Hidden assumption of the diagonal argument". January 25, 2016. https://www.academia.edu/20805963/Hidden_assumption_of_the_diagonal_argument (Kuan Peng identifies the problem as "the diagonal does not exist" because the array "is not square", but is unable to identify the logical flaw. This remains a goal in Kuan Peng's work at least as late as September 2022.)
- Hartl, Werner.(Dec. 16, 2020) "Cantors Diagonal Argument Fails." YouTube. (https://www.youtube.com/watch?v=3xlOzbzJXo8); updated March 3, 2023. (https://www.youtube.com/watch?v=135rdY7jHIA).
- Dauben, Joseph.(1979) Georg Cantor: His Mathematics and Philosophy of the Infinite. Harvard University Press. ISBN 0-674-34871-0
- Ferreirós, Jose.(2007) Labyrinth of Thought: A History of Set Theory and Its Role in Modern Mathematics. Birkhäuser.
- Sheppard, Barnaby.(2014) The Logic of Infinity. Cambridge University Press. Preprint -September 16, 2025